Conformal Killing gravity in static spherically-symmetric spacetimes

TL;DR

This paper introduces a second-order conformal Killing gravity framework for static spherically symmetric spacetimes, simplifying previous third-order equations and enabling analytic solutions for various physical scenarios.

Contribution

It establishes a second-order formulation of conformal Killing gravity equations, proving their equivalence to previous third-order equations and simplifying the derivation of static spherical solutions.

Findings

Derived analytic solutions for vacuum and linear electrodynamics.

Reobtained covariant static spherical solutions with simplified methods.

Identified compatible structures with anisotropic fluids and nonlinear electrodynamics.

Abstract

We identify an anisotropic divergence-free conformal Killing tensor $K_{jl}$ for static spherically symmetric spacetimes, and write the conformal Killing gravity equations as Einstein equations augmented by this tensor. The field equations are of second order: this fact allows for analytic solutions and considerably simplifies the derivation of results of previous studies based on the original Harada equations. In particular, we prove the equivalence of the known third order field equations, with the second order ones obtained by us in the conformal Killing parametrization. The structure of the Ricci tensor and of the conformal Killing tensor are compatible with both anisotropic fluid sources and (non)-linear electrodynamics. We reobtain covariantly and in simple steps the general static spherical solutions for vacuum and linear electrodynamics. Moreover we recover the purely magnetic Lagrangian functions that induce metrics of interest for black holes.